High-cycle fatigue of micron-scale polycrystalline silicon films ...

International Journal of Fracture 119/120: 449–474, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

High-cycle fatigue of micron-scale polycrystalline silicon films:fracture mechanics analyses of the role of the silica/siliconinterface

C.L. MUHLSTEIN∗ and R.O. RITCHIE∗Department of Materials Science and Engineering and the Materials Research Institute, The Pennsylvania StateUniversity, University Park, PA 16802 (E-mails: [email protected]; [email protected])

Abstract. It is known that micron-scale polycrystalline silicon thin films can fail in room air under high fre-quency (40kHz) cyclic loading at fully-reversed stress amplitudes as low as half the fracture strength, with fatiguelives in excess of 1011 cycles. This behavior has been attributed to the sequential oxidation of the silicon andenvironmentally-assisted crack growth solely within the SiO2 surface layer. This ‘reaction-layer fatigue’ mech-anism is only significant in thin films where the critical crack size for catastrophic failure can be reached by acrack growing within the oxide layer. In this study, the importance of the bimaterial (e.g., Si/SiO2) interface toreaction-layer fatigue is investigated, and the critical geometry and stress ranges where the mechanism is a viablefailure mode are established.

Key words: Fatigue, MEMS, silicon, thin films.

1. Introduction

Although conventional theories would suggest that silicon should not fatigue at room temper-ature, several studies have now confirmed that micron-scale mono and polycrystalline siliconstructural films, as used in microelectromechanical systems (MEMS), are susceptible to pre-mature failure in cyclic fatigue in ambient atmospheres (Allameh et al., 2001; Bagdahn andSharpe, 2002; Brown et al., 1997; Connally and Brown, 1992; Kahn et al., 1999; Kapels et al.,1999; Komai et al., 1998; Muhlstein et al., 2001a, b, 2002a, b; Tabib-Azar et al., 1992; VanArsdell and Brown, 1999). This research, which has often involved electrostatically-actuated,notched, cantilever-beam fatigue test structures (Figure 1) to measure the stress-life (S/N)fatigue curves (Muhlstein et al., 2001a, b), has revealed that in room-temperature air, thin-film silicon (∼ 2 µm thick) can exhibit ‘metal-like’ fatigue behavior with failures occurringat fully-reversed stress amplitudes of roughly one-half of the (single-cycle) fracture strength,after lives in excess of 1011 cycles (Figure 2). Recent studies by the authors have establishedthat the mechanisms associated with such fatigue failures can be ascribed to a process of‘reaction-layer fatigue’ involving the initiation, subcritical growth and catastrophic failureof nanoscale cracks within the native oxide layer (Figure 3) (Muhlstein et al., 2002a, b).Specifically, high-voltage transmission electron microscopy coupled with in situ monitoringof the compliance of the test structure to deduce crack sizes and oxide layer thicknesses, haveshown that the native SiO2 that initially forms on the silicon surface upon exposure to air(with a thickness and composition dictated by the environment and processing history) tendsto thicken in the high stress regions (e.g., at the notch) during subsequent fatigue loading. This

450 C.L. Muhlstein and R.O. Ritchie

Figure 1. Scanning electron micrograph of the micron-scale fatigue characterization structure and the notchedcantilever-beam specimen. The (a) toroidal mass and (b) notched cantilever-beam specimen (inset) are shown.

location then becomes the site for nanoscale cracks to initiate and grow due to the susceptib-ility of the silica to moisture-induced cracking1 . Similarly, preexisting crack-like faults in theSiO2 reaction layer may also grow during cyclic loading. The process then repeats itself untilthe critical crack size for failure of the entire test structure is reached, whereupon the siliconitself fractures catastrophically by transgranular cleavage (Muhlstein et al., 2002a, 2b). Therate-dependence of thin-film silicon fatigue failures is thus dictated by the cycle-dependentoxide thickening process and the time-dependent, environmentally-assisted, crack initiationand subcritical growth in the oxide layer.

The rate dependence of the reaction-layer fatigue process is invariably linked to the coupledinteraction of the oxide layer growth and cracking within it. When the growth rates of thesubcritical nanoscale cracks within the amorphous oxide layer were monitored (from thechange in specimen compliance), they were found to be decreasing throughout most of thefatigue life (Muhlstein et al., 2002a, b). Such behavior was ascribed to several factors, mostnotably the likely presence of compressive residual stresses in the oxide layer and the factthat the cracks were propagating in a layered SiO2/Si structure and approaching the SiO2/Siinterface. While the importance of residual stresses in understanding the driving force forcrack advance is clear, the role of the bimaterial interface is less obvious. The presence ofthe compliant SiO2 surface layer on the relatively stiff silicon structural film (polycrystallinesilicon at room temperature has an elastic modulus nearly three times that of amorphous SiO2)is known to lead to decreasing crack-driving forces that approach zero as the crack reaches theinterface (Beuth, 1992). However, a more detailed fracture mechanics analysis is necessary tounderstand the importance of this bimaterial interface and the change in the local geometryinduced by stress-assisted oxidation at the root of the notch to the evolution of fatigue damage1In the absence of the native oxide layer, i.e., through the use of alkene-based monolayer coatings on the siliconsurface, the susceptibility of thin-film silicon to cyclic fatigue failure has been shown to be markedly diminished(Muhlstein et al., 2002a, b).

High-cycle fatigue of micron-scale polycrystalline silicon films 451

Figure 2. Stress-life (S/N) fatigue behavior of 2-µm thick polycrystalline silicon thin films at ∼ 40 kHz inlaboratory air (Muhlstein et al., 2001a), determined using the thin-film fatigue characterization structure shown inFigure 1.

in polycrystalline silicon structural films. More importantly, materials other than silicon maybe prone to reaction-layer fatigue if they form a surface reaction-layer (upon exposure to ser-vice or manufacturing environments) that is susceptible to environmental- or cycle-dependentcracking. Specifically, the reaction-layer fatigue mechanism can lead to delayed failure of thinfilms of materials that are ostensibly immune to stress-corrosion cracking and fatigue in theirbulk form.

Accordingly, the objective of the present work is to establish the general requirements forreaction-layer fatigue and to apply this framework to establish the conditions for potentialfatigue failure of polycrystalline silicon thin films. However, before establishing the designmethodology, we first review some of the fundamental aspects of interfacial fracture mechan-ics, and then apply these elements to develop a general analytical approach for defining theregimes of susceptibility.

2. Interfacial fracture mechanics

The reaction-layer fatigue mechanism for silicon thin films suggests a configuration thatinvolves cracks oriented normal to a bimaterial interface, within a surface layer, as shownschematically in the inset in Figure 4. The linear-elastic stress-intensity factor, K, for this


Figure 3. Schematic illustrations of the reaction-layer fatigue mechanism at the notch of the polycrystalline siliconcantilever beam. (a) Reaction layer (native oxide) on surface of the silicon. (b) Localized oxide thickening atthe notch root. (c) environmentally-assisted crack initiation in the native oxide at the notch root. (d) Additionalthickening and cracking of reaction layer. (e) Unstable crack growth in the silicon film. The thickness of the oxide,h, and relative crack size (a/h) at each stage of the reaction are shown.

configuration has been derived for cracks loaded in tension by the plane strain, biaxial stressesthat can develop due to the mismatch in coefficient of thermal expansion between the layer andthe substrate (Beuth, 1992; Gecit, 1979). Such interfacial solutions must be bounded at oneextreme by the case of a surface crack of length, a, in a semi-infinite sheet, i.e., as a/h → 0,k → 1.1215σ

√πa, where h is the layer thickness and σ is the applied stress. However, when

the crack is near the interface (i.e., as a/h → 1), the typical 1/√

r stress singularity is notobserved and the crack-tip stress field, σij , is given by (Zak and Williams, 1963):

σij ∼ K̃r−sfij (θ), (1)

where K̃ is analogous to the stress-intensity factor (with an appropriate change in units), r isthe radial distance from the crack tip, fij (θ) is the angular distribution function, and s is theZak–Williams singularity (Zak and Williams, 1963). In the absence of an elastic mismatch,s = 1

2 , as expected from monolithic fracture mechanics (see Anderson, 1995; Broek, 1986).However, the presence of a bimaterial interface can have a marked effect on the singularityand the associated crack-tip stress field. The strength of the Zak–Williams singularity is givenby the root of:

cos(sπ) − 2α − β

1 − β(1 − s)2 + α − β2

1 − β2= 0, (2)

where α and β are the well-known Dundurs (1969) parameters, defined as:


Figure 4. Schematic of the edge-crack in a surface loaded in tension and the general features of the solutions forthe stress-intensity factor for a crack within a layer oriented normal to a bimaterial interface based on the generalsolutions, derived by Beuth (Beuth, 1992).

α = (E1 − E2)/(E1 + E2),

β = µ1(1 − 2ν2) − µ2(1 − 2ν1)

2µ1(1 − ν2) + 2µ2(1 − ν1).

(3)

In Equation (3), Ei ≡ Ei/(1 − ν2i ) in plane strain and Ei ≡ Ei in plane stress, where Ei, µi

and νi are, respectively, the Young’s modulus, shear modulus and Poisson’s ratio. The surfacelayer and the underlying substrate are denoted by the subscripts 1 and 2, respectively.

Consequently, the influence of the bimaterial geometry on the stress-intensity factor solu-tions for reaction-layer cracks enters calculations through the Dundurs parameters. The lim-iting cases of a short crack and a crack tip on the bimaterial interface have been analyzed byBeuth, who proposed the following general functional form for cracks contained within thesurface layer (Beuth, 1992):

KI ≈ σ√

πh

(1.1215

(a

h

)1/2 (1 − a

h

)(1/2)S(

1 + λa

h

)), (4)

where λ is a curve-fitting parameter.The general features of the linear-elastic stress-intensity factor solutions for cracks normal

to an interface, within a surface layer, are shown in Figure 4. It is readily apparent that the


Figure 5. (a) Monolithic case: nominal dimensions and configuration of the notched cantilever-beam specimenin the electrostatically-actuation fatigue characterization structure used to evaluate the S/N fatigue behavior ofpolycrystalline silicon. (b) Bimaterial case: configuration of the initial thickness of the native oxide, ho, notch rootoxide thickness, h, and crack length, a, used in the oxidized structure models. The thickness of the oxide layer onthe sides and root of the notch is exaggerated for clarity.

bimaterial case is a significant departure from the solutions for a monolithic (e.g., all silicon)structure:

K = σ√

πaf (a/W), (5)

where W is the width of the specimen, as shown in the inset in Figure 4. The elastic mismatchbetween the layer and substrate has a profound effect on the crack stability and driving force.For the case of a rigid film on a compliant substrate (i.e., α > 0), the crack-driving forceincreases without bound as the crack approaches the interface; similarly, the driving force iscontinuously increasing when there is no elastic mismatch between the layers (i.e., α = 0).However, for the case of a compliant film on a rigid substrate (i.e., α < 0), the driving forcefor the crack displays a maximum at an intermediate value of normalized crack length, (a/

h)max, after which the stress intensity decreases as the crack approaches the interface. Thesetrends are observed over the range of β encountered in typical structural materials.

The stability of cracks within the layer can be determined using conventional fracturemechanics principles. One type of sample geometry that is capable of exploring these prin-ciples is seen in Figure 5. In the absence of resistance-curve behavior, a crack will begin topropagate when K ≥ Kc and will be unstable if dK/d(a/W) > 0 (Anderson, 1995; Broek,1986). Analogous stability criteria can also be defined for fatigue-crack growth in terms of thestress-intensity range, �K. However, developing an unstable crack within the surface layer isa necessary, but not sufficient, requirement for catastrophic failure of the structure. In order to


Figure 6. Linear-elastic conditions for the penetration or deflection of a crack normal to a bimaterial interface(after (He and Hutchinson, 1989)). The ratio of the fracture toughness of the interface, Gc,interf to the fracturetoughness of the underlying substrate, Gc,2, is shown as a function of the elastic mismatch Dundurs parameter,α. Since the behavior is a weak function of β, only the β = 0 case is shown. For normal impingement, when theinterfacial fracture toughness is less than ∼ 1

4 that of the underlying substrate, the crack will be deflected by theinterface.

predict catastrophic failure of a structure due to cracks normal to a bimaterial interface con-tained completely within the surface layer, it is necessary to consider the fracture toughness ofthe layer, Kc,1, interface, Kc,int and the underlying substrate, Kc,2. Over-driven cracks in thelayer may arrest at, be deflected at, or penetrate the interface depending on the relative fracturetoughness of the substrate and the interface. The linear-elastic conditions for penetration anddeflection of a crack at a bimaterial interface have been defined by He and Hutchinson (Heand Hutchinson, 1989). In general, if the fracture toughness of the interface, expressed interms of the strain-energy release rate G, is less than ∼ 1

4 of that of the underlying substrate,a crack normal to the interface will be deflected by that interface (Figure 6). Therefore, over-driven cracks will tend to penetrate the bimaterial interface and the conditions for catastrophicfailure can be determined from the fracture toughness of the substrate except for the case ofvery weak interfaces. These general concepts of interfacial fracture mechanics, combined withconventional fracture mechanics for monolithic materials, can be used to develop a generalframework for understanding the conditions required for reaction-layer fatigue.


3. Analysis methodology

In order for reaction-layer fatigue to occur, a specific set of conditions must prevail in thematerial systems as well as the particular geometry of the specimen/component. Cracks con-tained within the surface layer that reach a critical size must penetrate the bimaterial interfaceand continue to propagate. The initial phase of the analysis is to apply monolithic fracture-mechanics solutions to establish the critical crack sizes associated with the materials, ap-plied stresses, and geometry of interest. If the critical crack sizes are commensurate with thereaction-layer thickness and the layer is susceptible to environmental- or cycle-assisted crackgrowth, further analysis using interfacial fracture mechanics solutions are necessary to assessthe risk of failure due to reaction-layer fatigue. In order for catastrophic failure to occur dueto a crack within the layer, the fracture toughness of the surface layer must be greater than,or (nearly) equal to, that of the substrate. Furthermore, the fracture toughness of the interfacemust be greater than some fraction of the substrate fracture toughness to insure that the crackwill penetrate the interface (Figure 6). If these requirements are met for material combinationswhere α > 0, cracks that initiate and grow within the surface layer will always reach a criticalcrack size and lead to catastrophic failure of the structure. However, if α < 0, the presenceof the intermediate maximum and subsequent decline in the stress-intensity factor leads totwo possible outcomes. If the reaction layer is sufficiently thick and the applied stress largeenough, a critical crack may develop. However, if the stress and/or layer thickness are toosmall, configurations can exist where cracks never reach a critical size within the reactionlayer. The amorphous SiO2 reaction layer on polycrystalline silicon substrates encountered inmicroelectromechanical systems falls into this latter category and will be explored at lengthin the following sections.

4. Numerical methods

The notched cantilever beams, used for thin-film fatigue testing at ∼ 40 kHz (Figure 1), were∼ 40 µm long, 19.5 µm wide, and 2 µm thick. The notch was located 9.8 µm from the baseand was 13 µm deep, with a root radius of approximately 1 µm, the smallest root radiusthat could be created given the processing conditions (Figure 5). The toroidal plate spanning60◦ (inner radius, rinner = 30 µm, outer radius, router = 300 µm) rotates about the midpointof the remaining ligament of the notched beam and serves as a mass to lower the naturalfrequency of the structure. A series of planar, finite element models of this structure wereevaluated using a commercial software package (ANSYS v. 5.7). Quasi-static finite elementmodels of the cracked, monolithic, notched cantilever beam were constructed and analyzed todetermine the stress-intensity factor, K, and the nondimensional compliance, CEt (where C

is the compliance, and t is the specimen thickness). In contrast to previous work (Muhlsteinet al., 2003), these models focused on the extremely short cracks encountered in the stress-lifefatigue behavior of Silicon films, i.e., for a/(W − D) << 0.01, where W is the width ofthe beam and D is the depth of the notch. Quasi-static analyses of cracked, oxidized siliconstructures were also evaluated to determine the stress-intensity factor for a crack containedwithin the reaction layer. Modal analyses of both cracked and flaw-free, oxidized structureswere used to establish the correlation between the natural frequency of the structure, whichgives the compliance of the structure, and the evolution of fatigue damage in the form ofnotch-root oxidation and cracking of this oxide.


4.1. MONOLITHIC SOLUTIONS

In order to establish the range of applied stresses and geometries where reaction-layer fatigueis an important mechanism, fracture-mechanics solutions of the monolithic fatigue charac-terization structure may be used. It is appreciated that the actual case is bimaterial structureinvolving the cracking of a very thin reaction layer on top of a much larger and stiffer substrate,but the monolithic solutions do provide some useful insight.

The structure was modeled as an elastically isotropic, notched, cantilever beam and massusing six-node, triangular, plane-strain, solid elements with unit thickness. Small elastic de-formations were assumed. The displacements of nodes located at the base of the notchedcantilever beam were set to zero to achieve the ‘built-in’ boundary conditions of the structure.An average of the Voigt and Reuss bounds for a random, polycrystalline silicon aggregate(Young’s modulus, E = 163 GPa, Poisson’s ratio, ν = 0.23 (Simmons and Wang, 1971))was used for the elastic properties of the polycrystalline silicon elements. The density of thepolycrystalline silicon, ρ in the modal analysis was assumed to be the same as the bulk, singlecrystal form (ρ = 2329 kg m−3 (Tatsumi and Ohsake, 1988)). The geometry of the mesh in thenotch region was selected so that a direct comparison between the monolithic and bimaterialmodels was possible. The size of elements at the notch root and in the rest of the structure wasrefined until the solutions to the models converged. A subspace extraction method was usedto determine the eigenvalues and eigenvectors for the system, establishing the first in-planebending frequency and mode shape of the structure.

Modal and quasi-static structural analyses of cracked, notched, cantilever-beam structureswere performed using the same material properties and finite element package as the crack-free structures using the geometry defined in Figure 5a. Akin to the uncracked structures,small deformations and elastic isotropy were assumed, displacements of nodes at the base ofthe cantilever structure were set to zero, and the material properties of an ideal polycrystallinesilicon aggregate were used. The model of the characterization structure, with a crack locatedon the centerline of the notch (Figure 5a), was constructed with six-node, triangular, plane-strain elements. The singularity at the crack tip was modeled with twelve circumferential,unskewed elements. The size of the elements was 1/20 of the crack length or smaller. The sizeof the remaining elements was refined until the solutions to the model converged. As withthe uncracked structure, a subspace extraction method was used to determine the eigenvaluesand eigenvectors for the system, establishing the first in-plane natural frequency and modeshape of the structure. A unit force was applied at the center of mass of the plate in a directionperpendicular to the long axis of the beam and the stress intensity was then calculated fromthe displacement field immediately behind the crack tip (Anderson, 1995; Broek, 1986). Thesize of the crack was gradually increased to establish the stress-intensity geometric function,f (a/(W − D)), for the cracked structure using standard fracture-mechanics procedures (An-derson, 1995; Broek, 1986). The stress-intensity factor and nondimensional compliance weredetermined for structures with normalized crack sizes from ∼ 0.0002 to 0.015, in order tofocus the analysis on the range of crack sizes relevant to reaction-layer fatigue. A nonlinear,least-squares regression scheme using a Levenberg–Marquardt algorithm was used to fit ageneral form of a geometric function for the stress-intensity factor to the numerical results.A root-finding algorithm employing a combination of bisection, secant, and inverse quadraticinterpolation methods (Forsythe et al., 1977) was used to establish the Zak–Williams singu-larity as well as critical values of crack size, oxide thickness, and other relevant parametersfrom the relationships developed from the finite element models and nonlinear regression.


4.2. INTERFACIAL SOLUTIONS

The oxidized fatigue characterization structure was modeled as an elastically isotropic,notched, bimaterial, cantilever beam and mass, with the geometry determined by scanningelectron microscopy (SEM) of released devices, as summarized in Figure 5b. The SiO2/Siinterface was placed 46% of the oxide layer thickness into the silicon structural film to modelthe loss of silicon to the reaction layer during the oxidation process (Jaeger, 1993). Furtheroxidation from the initial native oxide thickness of 30 nm (Muhlstein et al., 2002a, b) wasachieved by allowing the SiO2/Si interface at the root of the notch to progress into the siliconand the SiO2 outer surface to extend outward in a manner consistent with the consumptionof silicon by the oxidation reaction. Six-node, triangular, plane-strain, solid elements withunit thickness were used in the model and small elastic deformations were assumed. Thedisplacements of nodes located at the base of the notched cantilever beam were set to zero toachieve the ‘built-in’ boundary conditions of the structure. Akin to the monolithic structures,an average of the Voigt and Reuss bounds for a random, polycrystalline silicon aggregatewas used for the elastic properties of the polycrystalline silicon elements and the density ofthe polycrystalline silicon, ρ in the modal analysis was assumed to be the same as the bulk,single crystal form. The material properties of the oxide layer were assumed to be the sameas bulk, amorphous SiO2 (E ≈ 60 GPa, ν ≈ 0.2, ρ ≈ 2200 kg m−3). This bimaterial systemis described by Dundurs parameters of α = −0.47 and β = −0.18 and the correspondingZak–Williams singularity of s = 0.39. The size of elements within the oxide layer was setto 7.5% of the notch root layer thickness and the remaining elements were refined until thesolutions to the model converged. A subspace extraction method was used to determine theeigenvalues and eigenvectors for the system, establishing the first in-plane bending frequencyand mode shape of the oxidized structure.

Modal and quasi-static structural analyses of cracked, notched, bimaterial, cantilever-beamstructures were performed using the same material properties and finite element package asthe crack-free structures. As with the uncracked structures, small deformations and elasticisotropy were assumed, displacements of nodes at the base of the cantilever structure were setto zero, and the material properties of an ideal polycrystalline silicon aggregate and amorphousSiO2 layer were used. The model of the fatigue characterization structure with a crack withinthe oxide layer situated on the centerline of the notch (Figure 5b), was constructed with six-node, triangular, plane-strain elements. The singularity at the crack tip was modeled withtwelve circumferential, unskewed elements. The size of the elements was either 1/20 of thecrack length or 1/20 the remaining ligament, (h − a), whichever was smaller. The size of theremaining elements within the oxide layer was set to 7.5% of the notch-root layer thicknessand the rest of the mesh was refined until the solutions to the model converged. As withthe uncracked structure, a subspace extraction method was used to determine the eigenvaluesand eigenvectors for the system, establishing the first in-plane bending natural frequency andmode shape of the structure. A unit force was applied at the center of mass of the platein a direction perpendicular to the long axis of the beam and the stress intensity was thencalculated from the displacement field immediately behind the crack tip (Anderson, 1995;Broek, 1986). The size of the crack was gradually increased to establish the stress-intensitygeometric function, f (α, β, a, h), for the cracked structure using standard fracture-mechanicsprocedures (Anderson, 1995; Broek, 1986). A nonlinear, least-squares regression scheme us-ing a Levenberg–Marquardt algorithm was used to fit a general form of a geometric functionfor the stress-intensity factor to the numerical results. A root-finding algorithm employing a


combination of bisection, secant, and inverse quadratic interpolation methods (Forsythe et al.,1977) was used to establish the Zak–Williams singularity as well as critical values of cracksize, oxide thickness, and other relevant parameters from the relationships developed from thefinite element models and nonlinear regression. Finite element models of structures with notchroot oxides thicknesses, h, varying from 30 to 100 nm were evaluated for relative crack sizes,a/h, from 0 to 0.95; these values were selected based on previously published experimentalresults for polycrystalline silicon structural films (Muhlstein et al., 2002a, b).

5. Results and discussion

5.1. MONOLITHIC FRACTURE MECHANICS ANALYSES

We first consider the monolithic solutions and the critical crack sizes, ac, associated withcatastrophic failure of an edge-cracked sheet of polycrystalline silicon (Kc ∼ 1 MPa

√m)

loaded in tension during S/N testing. Using conventional stress-intensity solutions from theliterature (Tada et al., 2000), the computed values of ac are plotted as a function of the appliedstress, σapp, in Figure 7a for various widths W of the sheet. The stress-life fatigue data inFigure 2 would suggest that, for lives less than ∼ 1011 cycles, the relevant, fully reversedstress amplitudes are in the range of ∼ 2 to 5 GPa. For this range of stresses, it is apparent thatthe critical crack size for the micron-scale, thin-film silicon specimens is commensurate withthe largest, experimentally measured oxide layer thickness (h ∼ 100 nm) that forms undercyclic loading in ambient air (Muhlstein et al., 2002b). Since the oxide thickness is of theorder of the critical crack size (ac ∼ h) for such sheet widths, i.e., typical of thin-film silicon,reaction-layer fatigue can be considered as a viable failure mechanism.

The monolithic solutions (provided they have the appropriate accuracy at very small nor-malized crack lengths) can thus be used to evaluate the relevance of the surface-layer failuremechanisms over the stress ranges of interest. Furthermore, these solutions can provide insightas to why the effect of reaction-layer fatigue is essentially insignificant for bulk silicon, con-sistent with the fact that there have been no reported instances of bulk silicon being susceptibleto premature cyclic fatigue failures in ambient air environments. The calculations shown inFigure 7a can be represented in a more general form by normalizing the critical crack size bythe thickness of the native oxide, h, and the applied tensile stress normalized by the (single-cycle) strength of the material, σult, as shown in Figure 7b. For large specimens (e.g., W1 inFigure 7b), the critical crack size for failure at stresses below σult is much larger than thereaction-layer thickness. Reducing the size of the specimen, however, will eventually lead tocritical crack sizes that are similar in size to the native oxide (e.g., W < Wcrit in Figure 7b),as depicted schematically in Figure 8. Since for bulk silicon, ac >> h at applied stressesless than σult, cycle- and moisture-induced cracking of the oxide layer cannot lead to failureof the entire structure. While the plots in Figure 7 clarify the geometric requirements forreaction-layer fatigue, an alternative representation of the results provides additional insight.

A general failure map for susceptibility to reaction-layer fatigue can be generated byplotting the relative critical crack size, ac/h, for a given applied stress as a function of thespecimen size normalized by the thickness of the reaction layer, as shown in Figure 9. Thismap can be used to define the regimes where polycrystalline silicon may be subject to suchpremature failures:• At very low stresses, extraordinarily thick oxide layers would be required to support

a critical crack for failure. Since these thick layers cannot develop (even with stress-


Figure 7. (a) Monolithic fracture mechanics estimate of the critical crack size, ac , for a single-edge notchedtension (SENT) (Tada et al., 2000) as a function of applied stress, σapp, and specimen width, W , assuming afracture toughness, Kc, of 1 MPa

√m. The thickest, experimentally observed, native oxide in polycrystalline silicon

is indicated for comparison. (b) General representation of the effect of specimen size, W , on the relevance ofreaction-layer fatigue. For specimen sizes less than Wcrit, reaction-layer fatigue may be observed since ac is lessthan the notch-root oxide thickness, h.


Figure 8. Schematic of the relative crack size for (a) thin film reaction-layer fatigue and (b) bulk behavior forsilicon. When the critical crack size is of the same length scale as the oxide layer thickness, the fatigue mechanismis observed.

assisted oxidation), one would expect to see a threshold value below which reaction-layerfatigue should not be active.

• At intermediate stresses, two options appear to be possible. If the oxide layer is suffi-ciently thick, reaction-layer fatigue may occur. However, if the oxide thickness grows inservice or during testing (as has been observed (Muhlstein et al., 2002a, b), the monolithicsolutions suggest that increasing the reaction-layer thickness (while keeping the cracklength constant) lowers the driving force for crack advance by reducing the relative cracksize.2 This, as well as the fact that there may be cracks within the native oxide that arenot of a critical size, would suggest that subcritical crack growth should be possible andthat not all cracks are capable of propagating to failure.

• Finally, at very high stresses, the strength of the material is exceeded and simply cata-strophic failure occurs.

In previous studies of the notched cantilever-beam fatigue characterization structure, relat-ive crack sizes from ac/h ∼ 0 to 0.85 were evaluated using finite element methods (Muhlsteinet al., 2003). To facilitate a comparison with the interfacial models discussed below, additionalmonolithic fracture mechanics models of the characterization structure were evaluated in thepresent study for very short crack lengths, i.e., a/(W −D) 0.015. The calculated values ofthe stress-intensity factor were recast in the standard functional form, and the geometric func-2This conclusion is actually not correct and is an artifact of the monolithic solution, as shown in the followingsection.


Figure 9. Bounds for reaction-layer fatigue calculated using the monolithic fracture-mechanics solutions. Thecritical crack size is shown as a fraction of the reaction-layer thickness for a given specimen size as a function ofthe applied stress. Note that several regions are predicted: (i) catastrophic failure of the silicon, (ii) reaction-layerfatigue, (iii) no failure, and (iv) stable cracking in oxide (no failure).

tion for the structure, f (a/(W −D), shown in Figure 10, was fit to the following second-orderpolynomial expression:

f (a/(W − D)) = 1.9294 − 15.435 (a/(W − D)) + 66.657 (a/(W − D))2 . (5)

The nondimensional rotational compliance, CθEt , for this structure (Figure 11) was then fitto the following second-order polynomial expression:

CθEt = 4.2017 × 1011 + 7.3858 × 109 (a/(W − D)) + 9.2573 × 1012 (a/(W − D))2 . (6)

When these expressions are used to determine critical crack sizes over the range of stressesused during the stress-life fatigue tests (Figure 12), it is clear that they are again of the orderof the thickness of the native oxide.

5.2. INTERFACIAL FRACTURE MECHANICS ANALYSES

Using a bimaterial (amorphous SiO2-polycrystalline silicon) finite element model of the not-ched cantilever-beam fatigue characterization structure (Figure 5b), the linear-elastic stress-intensity solutions were calculated and used to determine the geometric function for theinterfacial crack as a function of crack size (Figure 13). It was found that the computed


Figure 10. Stress-intensity factor geometric function for very short cracks (a/(W − D) < 0.015) in the notchedcantilever-beam fatigue characterization structure.

mode II and III stress-intensity factors were orders of magnitude smaller than the mode Istress intensity. The geometric function for the stress-intensity factor, f (α, β, a, h), can beseen to exhibit an intermediate maximum

((a/h)max ≈ 0.8

)with increasing normalized crack

size (Figure 13b). Cursory examination of the geometric function in the neighborhood of thismaximum would (incorrectly) suggest that the influence of increasing oxide thickness on thestress-intensity factor changes depending on the value of normalized crack length. However,the

√h dependence shown in Equation (4) leads to an increasing stress-intensity factor with

increasing oxide thickness. Attempts to fit this stress-intensity function to the general form inEquation (4) using nonlinear regression resulted in a low quality curve fit due to the differencesbetween the geometry and loading conditions of the notched cantilever beam and the edge-cracked sheet analyzed by Beuth (1992). To improve the agreement between the fit functionand the numerical data, the following modified form of the geometric function proposed byBeuth was used:

f (α, β, a, h) = (mhh + bh)1.1215(a

h

)1/2 (1 − a

h

)(1/2)S(

1 + (mλh + bλ)a

h

), (7)

where mh, bh,mλ, and bλ are curve-fit parameters. Nonlinear regression established values of522244.2, 0.6389229, −1.043514 × 106 and −0.0268040 for mh, bh,mλ and bλ, respectively.

The interfacial K solutions for the notched cantilever-beam fatigue characterization struc-ture have direct implications for understanding the cracking behavior encountered in reaction-


Figure 11. Nondimensional compliance, CθEt , for very short cracks (a/(W − D) < 0.015) in the notchedcantilever-beam fatigue characterization structure.

layer fatigue. A comparison of the crack-tip driving force for a particular thickness of oxide(h = 100 nm) highlights the influence of the interface. As expected from the general trendsseen in the Beuth (1992) solutions, the presence of the interface between the compliant SiO2

layer on top of the polycrystalline silicon substrate lowers the driving force for crack advancefor a given applied stress and introduces an intermediate maximum in the variation of thestress-intensity factor with relative crack length (Figure 14).

An important aspect of the mechanics of the reaction-layer fatigue process is the relativeimportance of changes in crack length and oxide layer thickness on the driving force for crackadvance. The geometric function in Figure 13 establishes that increasing the relative cracksize leads to an intermediate peak in driving force that declines as the crack approaches theinterface. Consequently, the derivative of the stress-intensity factor with respect to the relativecrack size is positive, but decreasing, up to (a/h)max and becomes less than zero as the crackapproaches the interface (Figure 15a). What is less obvious is the impact of the increase in thethickness of the oxide on the driving force for crack advance. Monolithic solutions suggest thatincreasing the oxide layer thickness (while maintaining a constant crack length) reduces therelative crack size and, hence, should reduce the stress-intensity factor (Figure 9). However,this is not the case. From the interfacial solutions it is apparent that increasing the thicknessof the native oxide effectively pushes the influence of the bimaterial interface further awayfrom the crack tip, thereby raising the driving force for crack advance. Consequently, the


Figure 12. Estimated critical crack size, ac , for the notched cantilever-beam fatigue characterization structureover the stress range used in S/N fatigue tests of polycrystalline silicon assuming a fracture toughness, Kc, of1 MPa

√m.

growth of the oxide layer, which occurs in the highly stressed notch region during reaction-layer fatigue, always leads to an increase in the driving force for crack advance, as illustratedin Figure 15b, since the gradient of the stress-intensity factor with respect to the oxide layerthickness is always positive. This key feature of the reaction-layer process is not captured bythe monolithic K solutions.

In addition to reaction-layer fatigue, the presence of the intermediate maximum in thedriving force for crack advance in the interfacial K solutions also has other important im-plications for the cracking patterns observed in thin films and the stability of channelingcracks (Hutchinson and Suo, 1992; Ye et al., 1992). In the case of the reaction-layer fatiguemechanism, the interfacial effect of the compliant oxide layer on top of the relatively stiffpolycrystalline substrate sets discrete bounds for when the reaction-layer fatigue mechanismis expected to be active, i.e., when thin-film silicon may be susceptible to premature failurefrom this mechanism. Specifically, the crack driving force at (a/h)max represents the largeststress-intensity factor, Kmax, that can be generated for a given geometry and applied stresswhen the crack length is less than the layer thickness (i.e. a < h).

Combining this understanding of the role of the oxide layer thickness and the intermediatemaximum in the driving force for cracks within the layer provides a framework for definingwhen reaction-layer fatigue can occur in silicon. Figure 16 shows the effect of oxide layer


Figure 13. (a) The geometric function for the SiO2-Si bimaterial case of the notched cantilever beam fatiguecharacterization structure derived from finite element models. The local oxide thickness at the notch root, h, wasvaried over the experimentally observed range of native oxide thickness (30 to 100 nm). (b) Detail of the geometricfunction in the vicinity of the local maximum illustrating the change in effect of increasing oxide thickness.


Figure 14. Comparison of the monolithic and interfacial solutions for the linear-elastic stress-intensity factor, K ,for the notched cantilever-beam fatigue characterization structure. Note that (i) the presence of the amorphousSiO2 layer on top of the polycrystalline silicon leads to a marked decrease in the driving force for crack advanceunder the same applied remote stress, and (ii) the interfacial solution exhibits a local maximum at ∼ a/h = 0.8.

thickness and stress over the ranges observed experimentally in polycrystalline silicon. If thefracture toughness of the polycrystalline silicon (Kc ∼ 1 MPa

√m) and the environmentally-

assisted cracking threshold for silica glass (Kscc ∼ 0.25 MPa√

m) are superimposed on theplot, it becomes evident that there is a minimum oxide thickness that must be present inorder for cracks to initiate via environmentally-assisted cracking. What is less obvious is thatthese cracks can only reach a critical size (and cause failure of the entire structure) beforereaching the bimaterial interface if the oxide layer is sufficiently thick. In order for failure tooccur, the driving force for cracks growing within the native oxide must eventually exceedthe fracture toughness of the material. If the oxide layer is too thin, the fracture toughnessof the material cannot be reached for cracks within the surface layer of any size at stressesbelow the ultimate strength of the silicon. Given these solutions for the stress-intensity factorfor cracks in the oxidized, notched, cantilever-beam specimen, we can calculate the criticalvalue of oxide layer thickness required for crack initiation as a function of the applied stress(Figure 17). The critical initiation layer thickness, hi , of 2.9 nm must exist in order to reachKscc before the strength of the material is exceeded (∼ 5 GPa). Therefore, it is relatively easyfor cracks to be initiated in the reaction layers of thin-film silicon during fatigue since thenative oxides encountered in the heavily-doped structural film are quite thick. This is an


Figure 15. Effect of changes in oxide layer thickness, h, and the relative crack size, a/h, on the stress-intensityfactor, K , for the amorphous SiO2 on polycrystalline silicon notched cantilever-beam fatigue characterizationstructure. (a) The gradient of K with respect to the normalized crack length, a/h, shows that the driving force forcrack advance is increasing until an intermediate point after which the driving force for crack advance decreases.(b) The gradient of K with respect to the oxide thickness indicates that increasing the oxide layer thicknessincreases the driving force for crack advance.


Figure 16. The effect of amorphous oxide layer thickness, h, on the maximum possible stress-intensity factor,Kmax, for a given applied stress amplitude, σapp, in the polycrystalline silicon fatigue characterization structure.The critical initiation layer thickness, hi , the initial native oxide thickness, ho, the critical layer failure thickness,hc, and the maximum reaction layer thickness observed in polycrystalline silicon, hmax, are indicated on the plot.The threshold for environmentally-assisted cracking, Kscc, and the fracture toughness of the material, Kc, areshown for comparison.

interesting result since one might anticipate that steric hindrance (Michalske and Freiman,1983), where the crack would not be sufficiently open to allow for environmentally-assistedcracking, might be a limiting case. Both intrinsic flaws in the oxide layer and those initiatedby the synergistic interaction of the applied stress and the environment must be found within asufficiently thick oxide layer, otherwise the fracture toughness of the layer cannot be exceededand the component will never fail. For the polycrystalline silicon in our studies, a critical layerfailure thickness, hc, of 46.4 nm must be exceeded for a crack in the oxide layer to grow andexceed the fracture toughness of the material at stress levels below the strength of the material(∼ 5 GPa).

It should be noted that these critical values of layer thickness are conservative. Whether aparticular combination of applied stress and reaction-layer thickness could actually result infailure of the polycrystalline silicon should be determined from interfacial fracture-mechanicssolutions (Figure 17). Depending on the stress range, there are very wide ranges of reaction-layer thickness for which environmentally-assisted crack initiation cannot occur and wherecracks would never reach a critical size. This is presumably why the (relatively low stress)commercial applications of polycrystalline silicon MEMS are not currently plagued by fatigue


Figure 17. Critical oxide layer thickness for the reaction-layer fatigue mechanism. There are three regions asso-ciated with the reaction-layer fatigue mechanism. For a given maximum applied stress, there is a maximum oxidelayer thickness below which cracks cannot initiate via environmentally-assisted mechanisms. At larger oxide layerthicknesses where the maximum driving force for the given applied stress does not exceed the fracture toughnessof the material, cracks cannot propagate to failure. Finally, above a critical value of oxide layer thickness, reac-tion-layer fatigue failures can occur. The critical initiation layer thickness, hi , the initial native oxide thickness,ho, the critical layer failure thickness, hc , and the maximum oxide thickness observed experimentally, hmax, areindicated in the figure.

failures. However, this could well change when higher force actuators become available toMEMS designers.

5.3. MODAL ANALYSIS

Stress-life fatigue data can be useful for engineers designing structures and for researchers in-terested in comparing the relative susceptibility of materials and the efficacy of microstructuraland surface modifications. However, without fatigue-crack growth rate data, it is difficult to de-velop an understanding of the mechanisms of fatigue-crack initiation and growth. Compliancemeasurements are a convenient means of monitoring crack length during conventional crack-growth tests. In the case of micron-scale, resonant fatigue testing, the changes in specimencompliance manifest themselves as changes in the natural frequency of the specimen. Thechallenge lies in how to interpret these changes in specimen compliance.

Experimental evidence from (Muhlstein et al., 2002a, b) suggests that ‘damage’ in the formof stress-assisted oxidation as well as crack initiation and growth, occur in polycrystalline sil-


Figure 18. Three-dimensional surface representing the correlation between the specimen compliance, cracklength, and oxide thickness.

icon during the reaction-layer fatigue process. Also, it is known that both notch root oxidationand cracking result in similar changes in specimen compliance (Muhlstein et al., 2001a). Itis therefore necessary to have an understanding of the interrelationship between notch rootoxidation and cracking of the specimen. To determine this relationship, finite element modelsidentical to those used to determine the compliance and stress-intensity factor were evaluated.The resulting correlation between notch root oxidation and cracking and the compliance ofthe specimen is the surface shown in Figure 18. A given change in specimen compliance(i.e., the natural frequency) can be interpreted as a family of combinations of crack lengthand oxide layer thickness. The difficulty is to determine which geometry exists within thespecimen at a given point in time. Fortunately, we do have some information on the initial andfinal configurations of the system. At the start of the test, a native oxide some 30 nm thick ispresent, and at the end of the test the notch root oxide thickness can (in principle) be measuredfor each specimen. However, the trajectory between these two points across the surface inFigure 18 is still undefined. Although experimental data exist that contain information onthe crack and oxide layer growth rates, there is not sufficient information at present to usethe measured changes in specimen compliance to separate these two phenomena in order todefinitively establish fatigue-crack growth rates. Since the oxidation and cracking processesare likely coupled phenomena, this will probably require an independent measure of the cracklength or the oxide thickness throughout the test to interpret the changes in compliance of theresonant fatigue characterization structure.


6. Summary and conclusions

Although it is generally accepted that micron-scale polycrystalline silicon thin structural filmsare susceptible to fatigue failure in room temperature air, the failure mechanism has untilrecently remained a mystery. Converging evidence has suggested that the failure of suchsilicon films is due to a sequential process of stress-assisted oxidation and environmentally-assisted cracking of the native oxide layer, a process that is termed reaction-layer fatigue.Since it is apparent that the entire process of crack initiation and crack growth up to theonset of catastrophic failure occurs within this oxide layer, and not the silicon itself, in-terfacial fracture-mechanics solutions defining the variation in crack-driving force, e.g., thestress-intensity factor, with change in crack size provide additional insight into the natureof this fatigue mechanism. There is a critical distinction between reaction-layer fatigue andenvironmentally-assisted crack growth mechanisms such as film-induced cleavage (Newmanet al., 1989; Pugh, 1985) and slip dissolution (Ford and Silverman, 1979) where local crackadvance halts at or near the bimaterial interface and the critical crack size is large compared tothe reaction-layer thickness. Most importantly, the reaction-layer mechanism is one that canlead to delayed failure in thin films of materials that are ostensibly immune to stress-corrosioncracking and fatigue in their bulk form.

Based on fracture-mechanics solutions for the monolithic silicon and interfacial solutionsfor the SiO2/Si layered structure, both for standard geometries and for the notched cantilever-beam fatigue characterization structure used by the authors, the following specific conclusionsmay be made:1. Monolithic fracture-mechanics solutions can be used to establish when components are

small enough that the reaction-layer fatigue phenomenon is a viable failure mode. Micron-scale (as opposed to bulk) structures are susceptible to reaction-layer fatigue because thecritical crack size for failure is comparable to the reaction-layer thickness.

2. While monolithic fracture-mechanics solutions are useful for screening purposes, they failto accurately capture the importance of the thickening of the oxide layer during the testand its effect on the driving force for crack advance and final failure.

3. The thickness of the oxide layer is a critical parameter in the mechanics of the reaction-layer fatigue mechanism. In the present case of polysilicon, the native oxide at the rootof the notched cantilever-beam fatigue structure must exceed 2.9 nm in order to initiatecracks by a moisture-induced cracking mechanism and must exceed 46.4 nm to allowcatastrophic failure at stresses below the strength of the polycrystalline silicon. This im-plies that it is possible to initiate cracks within the oxide layer that cannot propagate tofailure, unless the oxide layer itself thickens throughout the test.

4. Interfacial fracture mechanics establish that a sufficiently thick reaction layer is requiredfor the fracture toughness of the material to be exceeded.

5. Oxidation and cracking of the reaction layer lead to similar changes in the compliance ofthe notched, cantilever-beam fatigue characterization structure; this severely complicatesthe quantification of the time-dependence of reaction-layer fatigue.

Acknowledgements

This work was funded by the Director, Office of Science, Office of Basic Energy Research,Division of Materials Sciences and Engineering of the U.S. Department of Energy under Con-tract No. DE-AC03-76SF00098. Additional funding for equipment was provided by the New


Energy and Industrial Technology Development Organization (NEDO), Tokyo, Japan. Theauthors wish to thank Dr S.B. Brown of Exponent, Inc., Natick, MA, for his initial support,Dr W. Van Arsdell for the original specimen design, and Dr Eric Stach and Prof. Roger Howefor helpful discussions.

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